Nonlocal Isoperimetric Inequality
For the nonlocal perimeter, there is also an isoperimetric inequality, and here the main hypothesis used on J is that it is radially nonincreasing.
Its proof uses the symmetric decreasing rearrangement, which replaces a given nonnegative function f by a radial function f∗. Let us recall briefly the definition and some basic properties of this rearrangement.
- 56.E.H. Lieb, M. Loss, Analysis. AMS Graduate Studies in Mathematics, vol. 14 (American Mathematical Society, Providence, 1987)Google Scholar